# When is Network Lasso Accurate?

## TL;DR

This paper establishes precise conditions on network structure and sampling for the accuracy of network Lasso in recovering graph signals, leveraging compressed sensing concepts.

## Contribution

It derives exact topological and sampling conditions that guarantee the accuracy of network Lasso for graph signal estimation.

## Key findings

- Conditions on network topology for accurate recovery
- Quantification of error based on node connectivity
- Extension of compressed sensing principles to network Lasso

## Abstract

The "least absolute shrinkage and selection operator" (Lasso) method has been adapted recently for networkstructured datasets. In particular, this network Lasso method allows to learn graph signals from a small number of noisy signal samples by using the total variation of a graph signal for regularization. While efficient and scalable implementations of the network Lasso are available, only little is known about the conditions on the underlying network structure which ensure network Lasso to be accurate. By leveraging concepts of compressed sensing, we address this gap and derive precise conditions on the underlying network topology and sampling set which guarantee the network Lasso for a particular loss function to deliver an accurate estimate of the entire underlying graph signal. We also quantify the error incurred by network Lasso in terms of two constants which reflect the connectivity of the sampled nodes.